Question

you stand 1.50 m in front of a wall and gaze downward at a small vertical mirror mounted on it. in this mirror you can see the reflection of your shoes. if your eyes are 1.85 m above your feet, through what angle should the mirror be tilted for you to see your eyes reflected in the mirror? (the location of the mirror remains the same, only its angle to the vertical is changed).

a. 89.4°
b. 31.7°
c. 126°
d. 54.2°
e. 63.3°

Explanation:

Step1: Consider the geometry of reflection

Let the distance from the person to the wall be $d = 1.50$ m and the height difference between the eyes and feet be $h=1.85$ m. For a ray of light to reflect from the eyes to the mirror and then to the eyes, we can use the law of reflection (angle of incidence equals angle of reflection) and basic trigonometry.
The angle of incidence $\theta_i$ and reflection $\theta_r$ are related to the geometry of the problem. If we consider the right - triangle formed by the person, the mirror, and the path of the light ray. The tangent of the angle related to the reflection geometry is given by $\tan\theta=\frac{h/2}{d}$.

Step2: Calculate the angle

We know that $\tan\theta=\frac{1.85/2}{1.50}=\frac{0.925}{1.50}\approx0.617$.
Then $\theta=\arctan(0.617)$.
$\theta\approx31.7^{\circ}$.

Answer:

B. 31.7°